IE tends to work with Report Manager, but most other browsers do not. Some browsers will show the Report Manager window but will not render a report.

Microsoft has no interest in making Report Services work well on anything other than Microsoft software, so if you want to use SSRS then your users need to be using IE. The 'best' version of IE to use is one that was new at about the same time that your version of SSRS was new. Don't expect all SSRS 2005 reports to render fully on IE10, and don't expect all SSRS 2012 reports to render fully on IE8, but most reports should be OK on most of the recent versions of IE.